The PID control scheme is named after its three correcting terms, whose sum constitutes the manipulated variable (MV). Hence:
Where Pout, Iout, and Dout are the contributions to the output from the PID controller from each of the three terms, as defined below.
The proportional term (sometimes called gain) makes a change to the output that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp, called the proportional gain.
The proportional term is given by:
where
Pout: Proportional term of output
Kp: Proportional gain, a tuning parameter SP: Setpoint, the desired value
PV: Process value (or process variable), the measured value e: Error = SP − PV
t: Time or instantaneous time (the present)
A high proportional gain results in a large change in the output for a given change in the error. If the proportional gain is too high, the system can become unstable (see the section on loop tuning). In contrast, a small gain results in a small output response to a large input error, and a less responsive (or sensitive) controller. If the proportional gain is too low, the control action may be too small when responding to system disturbances.
Droop
A pure proportional controller will not always settle at its target value, but may retain a steady-state error. Specifically, the process gain - drift in the absence of control, such as cooling of a furnace
towards room temperature, biases a pure proportional controller. If the process gain is down, as in cooling, then the bias will be below the set point, hence the term "droop".
Droop is proportional to process gain and inversely proportional to proportional gain. Specifically the steady-state error is given by:
e = G / Kp
Droop is an inherent defect of purely proportional control. Droop may be mitigated by adding a compensating bias term (setting the setpoint above the true desired value), or corrected by adding an integration term (in a PI or PID controller), which effectively computes a bias adaptively.
Despite droop, both tuning theory and industrial practice indicate that it is the proportional term that should contribute the bulk of the output change.
Integral term
The contribution from the integral term (sometimes called reset) is proportional to both the magnitude of the error and the duration of the error. Summing the instantaneous error over time (integrating the error) gives the accumulated offset that should have been corrected previously. The accumulated error is then multiplied by the integral gain and added to the controller output. The magnitude of the contribution of the integral term to the overall control action is determined by the integral gain, Ki.
The integral term is given by:
where
Iout: Integral term of output
Ki: Integral gain, a tuning parameter SP: Setpoint, the desired value
PV: Process value (or process variable), the measured value
e: Error = SP − PV
t: Time or instantaneous time (the present) τ: a dummy integration variable
The integral term (when added to the proportional term) accelerates the movement of the process towards setpoint and eliminates the residual steady-state error that occurs with a proportional only controller. However, since the integral term is responding to accumulated errors from the past, it can cause the present value to overshoot the setpoint value (cross over the setpoint and then create a deviation in the other direction). For further notes regarding integral gain tuning and controller stability, see the section on loop tuning.
Derivative term
The rate of change of the process error is calculated by determining the slope of the error over time (i.e., its first derivative with respect to time) and multiplying this rate of change by the derivative gain Kd. The magnitude of the contribution of the derivative term (sometimes called rate) to the overall control action is termed the derivative gain, Kd.
The derivative term is given by:
where
Dout: Derivative term of output
Kd: Derivative gain, a tuning parameter SP: Setpoint, the desired value
PV: Process value (or process variable), the measured value e: Error = SP − PV
t: Time or instantaneous time (the present)
The derivative term slows the rate of change of the controller output and this effect is most
noticeable close to the controller setpoint. Hence, derivative control is used to reduce the magnitude of the overshoot produced by the integral component and improve the combined controller-process stability. However, differentiation of a signal amplifies noise and thus this term in the controller is
highly sensitive to noise in the error term, and can cause a process to become unstable if the noise and the derivative gain are sufficiently large. Hence an approximation to a differentiator with a limited bandwidth is more commonly used. Such a circuit is known as a Phase-Lead compensator.
Summary
The proportional, integral, and derivative terms are summed to calculate the output of the PID controller. Defining u(t) as the controller output, the final form of the PID algorithm is:
where the tuning parameters are:
Proportional gain, Kp
Larger values typically mean faster response since the larger the error, the larger the
proportional term compensation. An excessively large proportional gain will lead to process instability and oscillation.
Integral gain, Ki
Larger values imply steady state errors are eliminated more quickly. The trade-off is larger overshoot: any negative error integrated during transient response must be integrated away by positive error before reaching steady state.
Derivative gain, Kd
Larger values decrease overshoot, but slow down transient response and may lead to instability due to signal noise amplification in the differentiation of the error.
Motion control
Motion control is a sub-field of automation, in which the position and/or velocity of machines are controlled using some type of device such as a hydraulic pump, linear actuator, or an electric motor, generally a servo. Motion control is an important part of robotics and CNC machine tools, however it is more complex than in the use of specialized machines, where the kinematics are usually simpler.
The latter is often called General Motion Control (GMC). Motion control is widely used in the packaging, printing, textile, semiconductor production, and assembly industries.
The basic architecture of a motion control system contains:
A motion controller to generate set points (the desired output or motion profile) and close a position and/or velocity feedback loop.
A drive or amplifier to transform the control signal from the motion controller into a higher power electrical current or voltage that is presented to the actuator. Newer "intelligent" drives can close the position and velocity loops internally, resulting in much more accurate control.
An actuator such as a hydraulic pump, air cylinder, linear actuator, or electric motor for output motion.
One or more feedback sensors such as optical encoders, resolvers or Hall effect devices to return the position and/or velocity of the actuator to the motion controller in order to close the position and/or velocity control loops.
Mechanical components to transform the motion of the actuator into the desired motion, including: gears, shafting, ball screw, belts, linkages, and linear and rotational bearings.
The interface between the motion controller and drives it controls is very critical when coordinated motion is required, as it must provide tight synchronization. Historically the only open interface was an analog signal, until open interfaces were developed that satisfied the requirements of coordinated motion control, the first being SERCOS in 1991 which is now enhanced to SERCOS III. Later interfaces capable of motion control include Profinet IRT, Ethernet Powerlink, and EtherCAT.
Common control functions include:
Velocity control.
Position (point-to-point) control: There are several methods for computing a motion trajectory. These are often based on the velocity profiles of a move such as a triangular profile, trapezoidal profile, or an S-curve profile.
Pressure or Force control.
Trans-mutational vector mapping.
Electronic gearing (or cam profiling): The position of a slave axis is mathematically linked to the position of a master axis. A good example of this would be in a system where two
rotating drums turn at a given ratio to each other. A more advanced case of electronic gearing is electronic camming. With electronic camming, a slave axis follows a profile that is a function of the master position. This profile need not be salted, but it must be an animated function.
Adaptive control
Adaptive control involves modifying the control law used by a controller to cope with the fact that the parameters of the system being controlled are slowly time-varying or uncertain. For example, as an aircraft flies, its mass will slowly decrease as a result of fuel consumption; we need a control law that adapts itself to such changing conditions. Adaptive control is different from robust control in the sense that it does not need a priori information about the bounds on these uncertain or time-varying parameters; robust control guarantees that if the changes are within given bounds the control law need not be changed, while adaptive control is precisely concerned with control law changes.